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        <title>A Brief Introduction to LRE</title>
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        <h1 align="center">An Introduction to LRE</h1>
		<p>Linear regression of efficiency (LRE) originated from recognition 
		that real-time qPCR could be greatly simplified through the application 
		of sigmoidal modelling, that in turn allows absolute qPCR to be 
		conducted without the need to construct standard curves. The following 
		provides an overview of how LRE was developed, based on the concepts 
		introduced by <a href="lre_literature.html">Rutledge and 
		Stewart (2008a)</a>.</p>
		<p>Real-time PCR was developed from the ability to monitor amplicon DNA quantity using some form of 
		fluorescence chemistry. This in turn generates an &quot;amplification 
		profile&quot; in which&nbsp;reaction fluorescence is 
		plotted against cycle number:</p>
		<p align="center"><b>An Amplification Profile</b><br>
		<img border="0" src="images/fc_plot1.gif" width="358" height="108"></p>
		<p>Each cycle is thus defined by the fluorescence signal it generates (F<sub>c</sub>), 
		which when using the fluorescent dye SYBR Green I, is directly proportional to 
		the amount amplicon DNA present at the end of the cycle. 
		Conventional qPCR methods determine target quantity based on the 
		relative position of a profile. A major limitation of this approach 
		(among many others) is that absolute quantification requires 
		construction of target-specific standard curves
		<a href="lre_literature.html">(Rutledge and Côté 2003)</a>. </p>
		<p>LRE expands the capabilities of qPCR by extending data analysis 
		beyond profile position. This was accomplished by 
		first determining the amplification efficiency produced during each 
		cycle within the central region of the profile, based on the increase in amplicon DNA relative to the amount 
		present at the beginning of the cycle. This of course is 
		the amount of amplicon present at the end of the previous cycle, referred to as F<sub>C-1</sub>:</p>
		<p align="center">
		<img border="0" src="images/lre_fa1.gif" width="187" height="58"></p>
		<p>This then generates a second parameter defining a cycle , which is 
		the 
		amplification efficiency it generated, referred to as &quot;cycle efficiency&quot; or E<sub>C</sub>. 
		Importantly, contrary to the historically held belief that PCR 
		amplification is an exponential process in which amplification 
		efficiency is constant, this revealed that amplification efficiency 
		progressively decreases as amplicon DNA increases, and that this 
		decrease is linearly coupled to amplicon DNA mass
		<a href="lre_literature.html">(Rutledge and Stewart 2008b)</a>.</p>
		<p>The core functionality of LRE was subsequently derived from the ability to view PCR 
		amplification not as the increase in amplicon DNA, but rather as the 
		loss in amplification efficiency. This is done by plotting the E<sub>C</sub> 
		of each cycle against its F<sub>C</sub>, generating what is referred to 
		as the &quot;LRE Plot&quot;:</p>
		<p align="center"><b>The LRE Plot<br>
		</b><img border="0" src="images/lre_plot1.gif" width="356" height="207"></p>
		<p>Although the 
		mechanism of this linear coupling is unknown, this provided the ability 
		to apply linear regression analysis to PCR amplification using&nbsp; the 
		equation:</p>
		<p align="center">
		<img border="0" src="images/LRE_linear_equation.gif" width="184" height="27"></p>
		<p align="left">
		Referred to as &quot;linear regression of efficiency&quot; or LRE, this generates values for amplification efficiency (E<sub>max</sub>), 
		determined from the Y-intercept, and the rate of loss in amplification 
		efficiency (ΔE), determined from the slope. Note also that the plateau phase 
		of a profile is defined as the maximal fluorescence (F<sub>max</sub>), 
		which corresponds to the X-intercept within the LRE plot, and can be 
		calculated using the equation:</p>
		<p align="center"><b>X-intercept<br>
		</b>
		<img border="0" src="images/Fmax_equation.gif" width="103" height="51"></p>
		<p align="left">
		The cycles included into the 
		linear regression analysis are taken from the central region of a profile, which avoids 
		distortions caused by low reaction fluorescence generated during early cycles, and aberrations associated with cycles within the upper 
		region. This central region is referred as the &quot;LRE 
		window&quot;, which is denoted by red circles within the LRE 
		and F<sub>C</sub> plots:</p>
		<p align="center"><b>LRE Analysis<br>
		</b><img border="0" src="images/lre_plot.gif" width="357" height="206"></p>
		<p align="center"><b>The LRE Window<br>
		</b><img border="0" src="images/fc_plot2.gif" width="355" height="107"></p>
		<p align="left">&nbsp;</p>
		<p align="left">In addition to generating a linear representation of PCR 
		amplification, adapting the classic Boltzmann sigmoid 
		function to PCR, generated a derivative that can be used to determine target quantity 
		directly from a cycle&#39;s F<sub>C</sub> reading: </p>
		<p align="center">
		<img border="0" src="images/Fo_equation.gif" width="230" height="73"></p>
		<p align="left">
		This generates the third parameter defining a cycle, which is the 
		predicted target quantity expressed in fluorescence unit, which is visually displayed in what is called the &quot;F<sub>0</sub> 
		Plot&quot;, with the cycles within the LRE window 
		denoted by red circles:</p>
		<p align="center"><b>The F<sub>0</sub> Plot<br>
		</b><img border="0" src="images/fo_plot.gif" width="331" height="108"></p>
		<p align="left">
		Target quantity is then derived by averaging the LRE window F<sub>0</sub> 
		values, which as described below, can be converted into the number of 
		target molecules by correlating fluorescence to DNA mass. </p>
		<p align="left">
		The fourth cycle parameter is the predicted F<sub>C</sub>, calculated using a second sigmoidal derivative:</p>
		<p align="center">
		<img border="0" src="images/LRE_model.gif" width="244" height="81"></p>
		<p align="left">
		When displayed in the F<sub>C</sub> plot as circles, this allows 
		assessment of the general conformity of a profile to the LRE model:</p>
		<p align="center"><b>Adding predicted F<sub>C</sub><br>
		</b><img border="0" src="images/fc_plot3.gif" width="355" height="108"></p>
		<p align="left">Among other attributes, this can reveal aberrant 
		amplification kinetics, such as &quot;plateau drift&quot;:</p>
		<p align="center"><b>An example of modest Plateau Drift<br>
		</b><img border="0" src="images/fc_plot4.gif" width="357" height="106"></p>
		<p align="left">As described in more detail in the
		<a href="../lre_window_selection/lre_window_selection_overview.html">LRE 
		Window Selection</a> section, it is important to exclude such aberrant 
		cycles from the LRE analysis, as they generate quantitative errors. </p>
		<p align="left">As described in the
		<a href="../optical_calibration/optical_calibration_overview.html">
		Optical Calibration</a> overview, the final step is to convert the average F<sub>0</sub> 
		into the number of target molecules, by first correlating DNA mass to 
		fluorescence units (FU). This is accomplished by amplifying a known 
		quantity of lambda gDNA from which an optical calibration factor 
		(OCF = FU / ng dsDNA) is determined. 
		Similar to a conventional fluorescence assay for quantifying DNA, 
		fluorescence is converted into DNA mass (M<sub>0</sub>), that is then 
		converted into the number of target molecules (N<sub>0</sub>) based on 
		amplicon size (A<sub>S</sub>):</p>
		<p align="center">
		<img border="0" src="images/fo_conversion.gif" width="325" height="59"></p>
		<p align="left">In summarize, LRE introduces a new paradigm for qPCR 
		based on deriving the target quantity directly from the&nbsp; 
		fluorescence readings within the central region of an amplification 
		profile. In combination with adopting lambda gDNA as a 
		universal quantitative standard, the need to construct target-specific 
		standard curves is eliminated, making it possible to conduct large-scale 
		absolute quantification with few resources beyond that needed for target 
		amplification. </p>
		<p align="left">See also:<br>
		<a href="../lre_window_selection/lre_window_selection_overview.html">LRE 
		window selection</a><br>
		<a href="../getting_started/implementing_lre_assays.html">Implementing 
		LRE-based assays</a></p>
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